The closest thing we have to a scientific answer to this question is the final section of Allan Williams's The Knight and the Blast Furnace. Williams runs a lot of tests and a lot of engineering/metallurgical calculations on medieval armour, medieval weapons and the materials they are made of. It culminates in Chapter 9.5, 'Did it work.'

The short answer is that it depends on which early matchlocks and at what ranges. As powder improved, matchlocks became much more effective - 'serpentine' powder was replaced by more effective 'corned' powder. Moreover, guns improved as well. The 16th century musket (introduced around the 2nd quarter of the century), a big, heavy gun, fired from a rest, was much more powerful than the lighter, earlier arquebus (which still served in numbers alongside the musket until it was replaced by other lighter guns like calivers). In short, good quality armour was effective at medium to long ranges against arquebuses using earlier types of powder, but was vulnerable to muskets, especially using later, more effective powder.

One English source of the Elizabethan era says that a contemporary heavy musket could:

-Kill a man in 'proof' armour at 100 yards

-Kill a man in 'common' armour at 400 yards

-Kill a man in no armour at 600 yards

Lighter guns like carbines, calivers and pistols would be much less effective.

Condensing William's own conclusions, here are the relevant stats, from pages 944-945:

Muzzle energies for various early firearms :

-early 15th c Hussite Handgun - Sepentine* Power - 500-1000 J

-16th c Arquebus - Sepentine powder - 1300 j

-16th c Arquebus - Corned powder - 1750 j

-16th c musket - Serpentine powder - 2300 J

-16th c musket - corned powder - 3000 J

These are -muzzle- energies, and a spherical projectile is not very efficient in terms of drag. So much of this power would be lost after say, 100 yards.

Now against these attacks, Williams identifies the following values for resistance from 2mm of modern mild steel plate, for various angles of attack, because it is harder to penetrate something from an angle. Straight-on hits against the curved surface of armour are very hard, so most hits would have been at around 30 degrees, at least:

straight on - 750 J

30 degree angle - 900 J

45 degree angle - 1050 J

If padding is worn under this, this could add 150 J to the energy required to penetrate it.

In addition, compared to modern mild steel plate, the materials of early modern armour would be more or less effective, depending upon carbon content, hardening, and slag content. The quaility of materials can be summarized as follows:

-Cheap iron - .5x as effective

-'normal' steel - .75x as effective

-'good' steel (somewhat hardened) - 1.1x as effective

-'best' steel (well hardened) - 1.5x as effective

If you glance at the stats above, early handguns would not be able to penetrate steel plate of the best quality at more than short range (remember, these are -muzzle- energies). But by the time corned powder comes into use, direct hits from short range by the strongest guns can penetrate even the best armour.

'Munition' armour of the sort that infantryman wore (made of poor iron or normal steel, occasionally, 'good' steel) would offer protection against arrows, pikes, and swords, but wouldn't be able to resist bullets at short to medium ranges.

You can run these numbers yourself if you like. Suppose a knight was wearing a breastplate of the finest quality, made by the workshop of Kolman Helmschmid in Augsburg. It is 2mm thick. The total resistance of this armour against incoming projectiles is as follows (using the multiplier above):

-Straight on - 1125 J

-30 degrees - 1350 J

-45 degrees - 1575 J

So only a point-blank, straight on hit from an early 16th century arquebus using serpentine powder would penetrate this breastplate. But a musket using corned powder would be able to penetrate this breastplate at up to say, medium ranges. Note that in the later 16th century breastplates become thicker. This increases the weight of armour, until in the 17th century more and more pieces of armour are discarded, until you are left with a very thick breastplate over a buff leather coat. However, at the same time, the quality of the steel was -declining- as armour became more 'brute force', which meant it had to be even thicker. Most 17th century armour is just wrought iron or low carbon steel. It is almost never heat treated to harden it.

TL/DR the Elizabethan gentleman quoted up top is probably right. But his great-granddad wouldn't have been able to say the same thing about an arquebus at the beginning of the century.

If you have access to an academic library that has the book, you can follow all the calculations yourself.

Matchlocks eventually became quite effective, certainly by the time of the Battle of Pavia ( 1525) they could be decisive. They required far less training than bows or crossbows, and though they might not penetrate heavy armor, most armies became far more lightly armored by the late 15th c. anyway- the Swiss halberdiers, like most, found mobility was far more important and wore little armor, if any.

But there's also the question of how effective the gunpowder was. It took quite some time to work out how to make something consistent and reliable; for example, to figure out that the best source for saltpeter was the earth under the outhouses of taverns that served wine, not beer. The very simple little, short medieval hand cannon, pointed more than aimed, were better at handling a propellant that might vary a lot in how fast it burned, how much gas it generated.

De Vries: Medieval Military Technology